Strongly Rational Sets for Normal-Form Games Gilles Grandjean

Ana Mauleony

Vincent Vannetelboschz

September 29, 2015

Abstract We introduce the concept of minimal strong curb sets which is a settheoretic coarsening of the notion of strong Nash equilibrium. Strong curb sets are product sets of pure strategies such that each player’s set of recommended strategies contains all actions she may rationally select in every coalition she might belong to, for any belief each coalition member may have that is consistent with the recommendations to the other players. Minimal strong curb sets are shown to exist and are compared with other well known solution concepts. We provide a dynamic learning process leading the players to play strategies from a minimal strong curb set only. Key words: Set-valued solution concept, Strong Nash equilibrium, Coalition, Strong curb set, Learning. JEL classi…cation: C70, C72.

y

CEREC, Saint-Louis University –Brussels, Belgium. E-mail: [email protected] CEREC, Saint-Louis University – Brussels; CORE, University of Louvain, Louvain-la-Neuve,

Belgium. E-mail: [email protected] z CORE, University of Louvain, Louvain-la-Neuve; CEREC, Saint-Louis University – Brussels, Belgium. E-mail: [email protected]

1

Introduction

Basu and Weibull (1991) have introduced a set-theoretic coarsening of the notion of strict Nash equilibrium: minimal curb (closed under rational behavior) sets. This set-valued solution concept combines a standard rationality condition, stating that the set of recommended strategies of each player contains all her best responses to whatever belief she may have that is consistent with the recommendations to the other players, with players’aim at simplicity, which encourages them to maintain a set of strategies as small as possible. We propose the concept of minimal strong curb sets which is a set-theoretic coarsening of the notion of strict strong Nash equilibrium.1 We require the sets to be immune not only against individual deviations, but also against group deviations. Strong curb sets are product sets of pure strategies such that each player’s set of recommended strategies contains all actions she may rationally select in every coalition she might belong to, for any belief each coalition member may have that is consistent with the recommendations to the other players. A strong curb set is minimal if it does not properly contain another strong curb set. Think of the set of recommendations to a player in a minimal strong curb set as a well-packed bag for a sports weekend: you may want to be prepared for di¤erent kinds of sports since you may like playing tennis with player 2 or playing golf with playing 3 or playing bridge with players 2, 3 and 4 or going alone for a jog. When a coalition forms, we assume that its members may talk to each other and commit to take a joint action. That is, we do not restrict the set of possible deviations by requiring that the deviation should be self-enforcing. By allowing for every possible deviations, our concept may be weak at determining actual play, but predicts with con…dence: if players have beliefs with support in a minimal strong curb set (say because they realize this is an equilibrium or because they are able to observe history of play to gain information on likely choices of others), then we can predict with con…dence that players will actually choose an action in the set. In Section 2 we recall notations and de…nitions. In Section 3 we formally de…ne the concept of minimal strong curb sets and we show that minimal strong curb sets exist in general. In Section 4 we compare minimal strong curb sets with strong Nash equilibria, coalition-proof Nash equilibria and (bayesian) coalitionally rationalizable 1

Many games of interest lack strong Nash equilibria (Aumann, 1959).

1

strategy pro…les. In Section 5 we provide a dynamic learning process where at each period players choose a best-response, possibly in groups, against beliefs formed on the basis of strategies used in the recent past. We show that if agents’ memory is long enough, play settles down in a minimal strong curb set. In Section 6 we conclude.

2

Preliminaries

Strict set inclusion is denoted by

and weak set inclusion is denoted by

. A

normal-form game is a tuple G = N; fAi gi2N ; fui gi2N , where N = f1; 2; : : : ; ng is

a …nite set of players, each player i 2 N has a nonempty, …nite set of pure strategies

(or actions) Ai and a von Neumann-Morgenstern utility function ui : A ! R, where A=

j2N Aj .

let X

i

=

The set of all games is denoted by . For every X =

j2N nfig Xj ,

i2N Xi

A,

8i 2 N . The subgame obtained from G by restricting the

action set of each player i 2 N to a subset Xi

Ai is denoted – with a minor

abuse of notation from restricting the domain of the utility functions ui to

j2N Xj

–by GX = N; fXi gi2N ; fui gi2N . The set of mixed strategies of player i 2 N with support in Xi

Ai is denoted by

(Xi ). Payo¤s are extended to mixed strategies

in the usual way. Beliefs are pro…les of mixed strategies. The pro…le of strategies where player i 2 N plays ai 2 Ai and the remaining agents play according to the mixed strategy pro…le each

= ( j )j2N nfig 2 (Aj ), BRi (

j2N nfig

(Aj ) is denoted (ai ;

i ).

For

ui (a0i ;

i)

2 Ai g is the set of pure best responses of player i against her belief

i.

i 2 N and a0i

i

i

2

j2N nfig

i)

= fai 2 Ai j ui (ai ;

i)

for

Basu and Weibull (1991) have introduced the concept of curb sets. A curb set is a product set X =

i2N Xi

where (a) for each i 2 N , Xi

set of pure strategies; (b) for each i 2 N and each belief

Ai is a nonempty i

of player i with

support in X i , the set Xi contains all best responses of player i against her belief: 8i 2 N , 8

i

2

j2N nfig

(Xj ), BRi (

i)

Xi . Curb sets are product sets of pure

strategies such that each player’s set of recommended strategies contains all bestreplies to whatever belief she may have that is consistent with the recommendations to the other players. Since the full strategy space is a curb set, particular attention is devoted to minimal curb sets. A curb set X is minimal if no curb set is a proper subset of X. Every game G possesses at least one minimal curb set (Basu and Weibull, 1991). 2

A strong Nash equilibrium is a strategy pro…le such that no coalition has a joint deviation that bene…ts all of them (Aumann, 1959). Coalitions are nonempty subsets of players (J such that J let X

J

j2N nJ Xj .

=

N and J 6= ;). For every X

according to the mixed strategy pro…le For every J

J ).

J i

denote by 2

i2N Ai

N,

The pro…le of strategies where players belonging to coalition

J play according to the strategy pro…le aJ 2 (aJ ,

A and J

J

N , i 2 J, and

the marginal distribution of

i2J Ai

and the remaining players play

= ( j )j2N nJ 2 i

= ( j )j2N nfig 2 i

over A

J.

(Aj ) is denoted

j2N nJ

j2N nfig

(Aj ), we

The strategy pro…le a

is a strong Nash equilibrium if and only if, for each a 6= a , there exists

some player i such that ai 6= ai and ui (a)

ui (a ). A strong Nash equilibrium is

strict if the last inequality holds strictly.

3

Strong curb sets

While the concept of curb sets is a set-theoretic coarsening of the notion of strict Nash equilibrium, we now introduce the concept of strong curb sets which is a settheoretic coarsening of the notion of strict strong Nash equilibrium. We require the set to be immune not only against individual deviations, but also against coalitional deviations. De…nition 1. For each pro…le of beliefs

=(

the set of coalitional best-responses of coalition J CBRJ ( ) = fbJ 2

i2J Ai

(ii) @ b0J 2

Given a pro…le of beliefs

j (i) 8i 2 J, ui (ai ; i2J Ai

i )i2N

with

i

N is i)

2 J i ),

ui (bJ ;

J i)

such that 8i 2 J, ui (bJ ;

j2N nfig

(Aj ),

8ai 2 Ai and

< ui (b0J ;

J i )g.

, a strategy pro…le bJ for coalition J is a coalitional

best-response if (i) each member i 2 J prefers to join coalition J and playing bJ

rather than playing her individual best-response against her belief other pro…le

b0J

6= bJ such that all members of J strictly prefer

b0J

i,

(ii) there is no

to bJ . Conditions

(i) and (ii) capture some rudimentary form of coalitional rationality. First, a sensible concept of coalitional rationality should prescribe coordination on strategy pro…les so that all coalition members have incentives to join the group. Second, it should be conceivable that members of coalition J will never coordinate their play on strategy pro…les that are Pareto dominated. Of course, CBRfig ( ) coincides with BRi ( 8i 2 N . 3

i)

Example 1. Consider the normal-form games G1 and G2 . L

R

L

R

U

4; 5

0; 0

U

2; 0 0; 0

D

0; 0

3; 2

D

0; 0 0; 2

G1

G2

Take the normal-form game G1 and let J = f1; 2g. Condition (i) makes that (U; R)

and (D; L) are never coalitional best-responses for J whatever

. Condition (ii)

makes that (D; R) is not a coalitional best-response for J whatever strategy pro…le (U; L) satis…es both conditions whatever

. However, the

. Thus, CBRf1;2g ( ) =

f(U; L)g. Notice that the set of coalitional best-responses, CBRJ ( ), may be empty if jJj with

2. Take the normal-form game G2 and consider the beliefs

1 (L)

= 1 and

2 (D)

1

= 1. Then, BR (

and the expected payo¤s are u1 (U;

1)

1)

=(

2

= fU g and BR (

= 2 and u2 (R;

2)

2)

1;

2)

= fRg

= 2. Thus, we have

that CBRf1;2g ( ) = ;. A set X is a strong curb set if the belief that only strategies in X are played implies that players and coalitions have no incentives to use other strategies than those belonging to X. De…nition 2. A strong curb set is a product set X = (a) for each i 2 N , Xi (b) for each J

where

Ai is a nonempty set of pure strategies;

N and each vector of beliefs

with each belief

i2N Xi

i

=(

1 ; :::;

N)

of the players

having support in X i , the product set XJ =

j2J Xj

contains all coalitional best-responses of coalition J against the beliefs of its members: 8J

N; 8

CBRJ ( )

=(

1 ; :::;

n)

with

i

2

l2N nfig

(Xl ), i 2 N ,

j2J Xj .

Strong curb sets are product sets of pure strategies such that each player’s set of recommended strategies contains all actions she may rationally select in every coalition she might belong to, for any belief each coalition member may have that is

4

consistent with the recommendations to the other players.2 Each coalition member is allowed to hold a di¤erent belief concerning the play of others in the set of recommended strategies to assess the pro…tability of the deviation. Thus, the coalition members may disagree on where the deviation leads to. In addition, players do not update their belief by trying to understand why some coalitional action is a bestresponse for the other players of the coalition. Rather, a deviation is deterred if, for each pro…le of beliefs of the coalition members, at least one player of the group is strictly better o¤ by blocking the deviation. Notice that each strong curb set is a curb set and hence contains the support of at least one Nash equilibrium in mixed strategies. A strong curb set X is minimal if no strong curb set is a proper subset of 3

Establishing existence of minimal strong curb sets in …nite games is simple.

X.

The entire pure-strategy space A is a strong curb set. Hence the collection of strong curb sets is nonempty, …nite (since A is …nite) and partially ordered by set inclusion. Thus, a minimal strong curb set exists.4 Proposition 1. Every normal-form game G has a minimal strong curb set. Proposition 2. If X is a minimal strong curb set of G = N; fAi gi2N ; fui gi2N ,

then X is a minimal strong curb set of the subgame GX = N; fXi gi2N ; fui gi2N . Proof. Let X be a minimal strong curb set of G and Y a strong curb set of G. Thus, there exists J with

i

2

j2N nfig

j 2 J. Having e aJ 2 Thus, e aJ 2

(Yj ) such that uj (e aJ , j2J Aj nXj

j2J Xj nYj ,

N, e aJ 2

J j)

X. It implies that Y is not j2J Aj nYj

uj (bj ;

j)

and

=(

i )i2N

for all bj 2 Aj , for all

would contradict that X is a strong curb set of G.

implying that Y is not a minimal strong curb set of GX .

Proposition 2 implies that it is not possible to re…ne the notion by iteratively

eliminating strategies that do not belong to a particular minimal strong curb set. 2

Players choose pure strategies and hold mixed beliefs. The notion of strong curb set can be

easily extended to mixed strategies simply by accommodating the de…nition of CBR. 3 The product set of actions chosen in every strict strong Nash equilibrium is a minimal strong curb set. Conversely, for every minimal strong curb set composed of one action per player, the strategy pro…le in which each player selects this action is a strict strong Nash equilibrium. 4 In the appendix we show that the existence result holds for every game G 2 G, where G is the class of normal-form games G = N; fAi gi2N ; fui gi2N

such that for each player i 2 N =

f1; 2; : : : ; ng, Ai is a compact subset of a metric space and ui : A ! R is a continuous von Neumann-Morgenstern utility function.

5

4

Relationships with other solution concepts

Minimal strong curb sets always exist and allow us to make reasonable predictions in games in which a strong Nash equilibrium does not exist. Example 2. Consider the normal-form game G3 . L

C

R

U

4; 4 0; 5

0; 0

M

0; 5 2; 2

0; 0

D

0; 0 0; 0 a; 1

For a

0, this game has two Nash equilibria: the pure strategy Nash equilib-

rium (D; R) and the mixed one ( (1=3; 2=3; 0). For 0

U;

M;

D)

= (1=4; 3=4; 0) and (

L;

C;

R)

=

a < 4 none of these Nash equilibria are immune to the joint

deviation (U; L), implying that the game G3 has no strong Nash equilibrium. The only minimal strong curb set of G3 is ffU; M g

fL; Cgg when a < 4. Indeed, for

beliefs with support in the set, each player’s individual best-responses lie in the set. In addition, player 2 blocks the joint deviation to (D; R) since she can be sure of having a payo¤ greater than 1 by playing in the set.

Even if a strong Nash equilibrium exists and is unique, it may not be the only valid prediction of the game. Consider again the game G3 for a > 4. The strategy pro…le (D; R) is the unique strict strong Nash equilibrium. The set composed of those actions is thus a minimal strong curb set. But, fU; M g

fL; Cg is another minimal

strong curb set since for beliefs with support in the set, player 2 has no incentives to choose an action outside the set, even jointly with player 1. Player 1 would be willing to jointly deviate to (D; R), but without the consent of player 2, she prefers not to play D. Coalition-proof Nash equilibria always exist for two player games (Bernheim, Peleg and Whinston, 1987). They consist of Nash equilibria that are not Pareto dominated by another Nash equilibrium. In the game G3 , the two Nash equilibria are coalition-proof when a > 4=3 while only the mixed Nash equilibrium is coalitionproof when a

4=3. For 4=3 < a < 4, (D; R) is a coalition-proof Nash equilibrium

but the only minimal strong curb set is fU; M g

fL; Cg. Thus, the support of a

coalition-proof Nash equilibrium is not necessarily contained in a minimal strong curb set. The following example reveals that the predictions obtained under the

6

minimal strong curb set may be more plausible than those obtained under the notion of coalition-proof Nash equilibrium. Example 3 (Ambrus, 2006). Consider the normal-form game G4 . L

C

U

2; 1; 0

0; 0; 0

9; 9; 9

M

2; 0; 1

1; 0; 2

D

9; 9; 9

9; 9; 9

R

L

C

U

1; 2; 0

0; 2; 1

9; 9; 9

9; 9; 9 M

0; 0; 0

0; 1; 2

9; 9; 9

9; 9; 9

D

9; 9; 9

9; 9; 9

l L

R

9; 9; 9

c

C

R

U

9; 9; 9

9; 9; 9

9; 9; 9

M

9; 9; 9

9; 9; 9

9; 9; 9

D

9; 9; 9

9; 9; 9

8; 8; 8

r The unique coalition-proof Nash equilibrium of G4 is (D; R; r), while the unique minimal strong curb set is ffU; M g

fL; Cg

fl; cgg.

Contrary to curb sets, strong curb sets may include strategies that are strictly dominated or even not rationalizable. In the Prisoner’s Dilemma, the action cooperate is strictly dominated but belongs to the unique minimal strong curb set. Ambrus (2006) and Luo and Yang (2009) have proposed the concepts of coalitional rationalizability and bayesian coalitional rationalizability, respectively.5 There is no relationship between minimal strong curb sets and (bayesian) coalitionally rationalizable sets. (Bayesian) coalitional rationalizability may have more cutting power than minimal strong curb sets, as in the Prisoner’s Dilemma. The converse may also be true. Minimal strong curb sets may have more cutting power than (bayesian) coalitional rationalizability as in Example 2 where for 4=3

a

4, every strategy

is (bayesian) coalitionally rationalizable while only a subset of those belong to the unique minimal strong curb set.

5

Learning to play minimal strong curb strategies

We now provide a class of dynamic learning processes where groups of players may coordinate their actions that leads the players to play strategies from a minimal 5

See also Ambrus (2009) and Herings, Mauleon and Vannetelbosch (2004).

7

strong curb set only. In line with Hurkens (1995), players observe actions played recently, form their beliefs upon these observations, and play best-responses to those beliefs. Suppose a game G = N; fAi gi2N ; fui gi2N is played once every period. The

players are partitioned into classes C1 ; C2 ; :::; Cn such that ui = uj and Ai = Aj

if i; j 2 Ck . In each period, one player is drawn at random from each of n dis-

joint classes C1 ; C2 ; :::; Cn , to play the game G in that period. These players are partitioned into coalitions to form a coalition structure J = (J1 ; J2 ; : : : ; JM ) such that Jk \ Jl = ; for k 6= l and [M k=1 Jk = N . Let J be the …nite set of coalition structures. In period t, the history ht = (at

K

; :::; at 1 ) is a description of how

the game has been played in the K previous periods, where ak 2 A is the action

pro…le chosen by the n players in period k. We de…ne the state space H = AK to consist of all histories h = (a K ; :::; a 1 ) of length K. Call b h 2 H a successor of h 2 H if b h is obtained from h by deleting the leftmost element and by adding

some element a 2 A to the right. Let r(b h) denote the rightmost element of b h 2 H.

For h = (a

K

; :::; a 1 ) 2 H, let

i (h; k)

= fai k ; :::; ai 1 g denote the set of strategies

played by player i in the k last periods, for k

K. We assume that the choice of

the players is time-independent. The learning process can thus be described by a stationary Markov chain on the state space H = AK . Let P : H H ! [0; 1] be a transition matrix, where P (h; b h) is the probability of moving from state h 2 H to state b h 2 H in one period and b P (h; b h) = 1 for all h 2 H. A learning process h2H

is described by a transition matrix P 2 P, where P is de…ned as follows.

De…nition 3. Let P be the set of transition matrices P that satisfy for all histories h, b h 2 H, P (h; b h) > 0 if and only if (i) b h is a successor of h, (ii) there exists some J 2 J and

=(

1 ; :::;

n)

with

i

2

j2N nfig

( j (h; K)) such that r(b h) = (aJ )J2J

with aJ 2 CBRJ ( ) if CBRJ ( ) 6= ? and aJ 2

i2J

BRi (

i)

otherwise.

At each period every player chooses an action. This action can be chosen individually or in group, and is chosen after having observed the recent past play. When a group of players coordinate their actions, they choose a Pareto undominated action pro…le such that each member of the group bene…ts from playing jointly. In state h, if coalition J

N has a coalitional best-response aJ 2 CBRJ ( ) given

a pro…le of beliefs with support in the set of strategies played in the recent past, then the process moves with positive probability from state h to some state b h in 8

which coalition J plays aJ . To determine the outcome of such learning processes, what matters is to identify the set of states that can be reached from a state h with positive probability and those that cannot be reached. Since the exact probability does not matter, we do not have to specify a particular process of belief formation nor a protocol of coalition formation. We only require that every such belief with support in the set of actions played recently and every partition of the players occur with positive probability. For each k 2 N, P k : H

H ! [0; 1] denotes the k-step transition probabilities

of the Markov process with transition matrix P 2 P : P 1 = P and P k = P P k 1 b for k > 1. We will write h h if there exists k 2 N satisfying P k (h; b h) > 0. Now de…nes a weak order on H. We can de…ne an equivalence relation on H: h b h ,h

b h and b h

h. Let [h] denote the equivalence class that contains h and let

Q = f[h] j h 2 Hg denote the set of equivalence classes. A partial order on Q is given by: [h] [b h] , b h h. The minimal elements with respect to the order

are called ergodic sets. The other elements are called transient sets. If the process leaves a transient set it can never return to that set. If the process is in an ergodic set it can never leave that set. The elements belonging to ergodic and transient sets are called ergodic and transient states, respectively. In any …nite Markov chain, no matter where the process starts, the probability that the process is in an ergodic state after k steps tends to 1 as k tends to in…nity. Proposition 3 states that if memory is long enough (K high enough), each ergodic set Z of a learning process with transition matrix P 2 P is composed of minimal strong curb elements only, i.e. Z is such that Z

X K for some minimal strong curb set X of G.

Proposition 3. There exists K 2 N such that for all …nite K

Markov chain with transition matrix P 2 P; if Z (a) Z (b) @Y

K and every

H is an ergodic set then

X K for some minimal strong curb set X. X such that Z

Y K for some minimal strong curb set X.

The following lemma will be useful to prove Proposition 3. Lemma 1. Let ht = (xK t ; :::; x1 ; a1 ; :::; at ) be a particular history. If the players draw their beliefs from

i2N

t i (h ; t)

then

(a) the process moves with positive probability to an history ht+1 such that i2N

t+1 ;t i (h

+ 1) if

i2N

t i (h ; t)

9

is not a strong curb set;

i2N

t i (h ; t)

(b) the process moves with probability 1 to an history ht+1 such that t+1

i (h

i2N

; t + 1) if

i2N

t

i (h

i2N

t i (h ; t)

=

; t) is a strong curb set.

Proof. Easy and therefore omitted. n i=1

Proof of Proposition 3. Let L = 1 and let K

K =L+M

jAi j

(n

n i=1

1). Let M =

K be …nite. Let P 2 P.

jAi j

n. Take

(a) We will show that (i) from any history h1 2 H, the process moves with

positive probability in L

1 steps to a state hL 2 H such that

L i (h ; L)

i2N

is a

L

strong curb set, (ii) from state h , the process moves with positive probability in M steps to a state hL+M 2 H such that

set, and (iii) if Z

i2N

L+M ; M) i (h

H is an ergodic set then Z

is a minimal strong curb

X K for some minimal strong curb

set X. (a.i) Let a1 ; :::; aT 2 A be such that at+1 2 =

1. By de…nition of L, we have T t+1

i (h

i2N

i2N

t i (h ; t)

i2N

1 i (h ; 1)

L since

for all t = 1; :::; T contains n actions,

; t + 1) contains at least one additional action than

i2N

n i=1 jAi j 1 1

the action space A, which is the largest strong curb set, contains times part (a) of Lemma 1, we have h1

and

i2N

i (h

; ) is a strong curb set.

hL = (xK

h

L

h

i2N L

1

of them.

1

L

i (h

h = (xK

; :::; x1 ; a1 ; :::; a )

From part (b) of Lemma 1, we have

; :::; x1 ; a1 ; :::; a ; :::; aL ) such that

(a.ii) Let X L+M

and

L such that, starting from h1 = (xK 1 ; :::; x ; a ) and ap-

Thus, there exists a plying

t i (h ; t)

i2N

L i (h ; L)

=

i2N

i (h

; ). L

; L) be a minimal strong curb set. We have h

M

= (:::; a ; :::; a ; b ; :::; b ) such that

i2N

L+M ; M) i (h

= X since every strat-

egy in a minimal strong curb set is an element of a coalitional best-response to some belief with support in the set. We then have hL+M such that

i2N

L+K

i (h

hL+K = (b1 ; :::; bM ; c1 ; :::; cK

M

)

; K) = X by application of part (b) of Lemma 1. Finally, for

all k 2 N, for all hL+K+k such that hL+K

hL+K+k , we have

i2N

L+K+k ; K) i (h

X by De…nition 3. The set X K thus contains an ergodic set. (a.iii) By contradiction, suppose there exists an ergodic set Z such that Z * X K for all minimal strong curb set X of G. Thus Z contains an ergodic state h 2 H

such that h 2 = X K for all minimal strong curb set X. Applying (a.i) and (a.ii), we

have h

h0 such that h0 2 Y K for some minimal strong curb set Y and there exists

an ergodic set W

Y K . We do not have h0

contradicting the fact that Z is an ergodic set.

h since h 2 = W by assumption,

(b) Follows directly from Proposition 2 and part (a) of Lemma 1. 10

It is shown in Proposition 3 that if memory is long enough, the probability that players are playing a minimal strong curb strategy pro…le after k steps of the learning process tends to 1 as k tends to in…nity. To prove this result, we show that from every state h, there exists a …nite sequence of steps that occurs with positive probability leading to a state h0 where the set of strategies used in the recent past span a minimal strong curb set. Once h0 is reached, the players draw their belief from the minimal strong curb set and thus keep best-responding in that set. Let us make some remarks about this result. First, notice that it may be that some strategy pro…les a 2 X are never played in any repetition of the game. In the Prisoner’s Dilemma, either both players cooperate or both defect in each period.

However, once the process has converged to a minimal strong curb set, the players will select in…nitely often each strategy belonging to this set. In other words, play will not settle down to a proper subset of a minimal strong curb set. Second, the lower bound K we use in the proof of Proposition 3 is not tight. It may well be that K < K and each ergodic set of the game are subset of minimal strong curb histories. Third, the time k needed to move in a minimal number of steps from a strong curb history ht to a history ht+k such that the strategies chosen in the k last period span a minimal strong curb set X is exactly k = max(jX1 j ; :::; jXn j) if every

action ai 2 Xi of each player i 2 N is an individual best-response to some belief in

the set. This time may increase if minimal strong curb actions only belong to some coalitional best-responses and the same player is involved in di¤erent coalitional

moves. The maximal number of period needed to realize the transition from ht to ht+k in a minimal number of steps is M =

n i=1

the following example.

jAi j

n. This is illustrated through

Example 4. Consider the normal form game G5 L

R

U

2; 2; 1

0; 3; 3

U

0; 1; 0 2; 0; 2

D

3; 0; 1

1; 1; 1 D

1; 1; 0 3; 0; 0

l

L

R

r

We have that the unique minimal strong curb set is ffU; Dg

M =

n i=1

jAi j

n = 3. Suppose the process is in state ht where t

A. Let k be the smallest integer such that h i2N

t+k ; k) i (h

fL; Rg

t+k

h

i2N

fl; rgg and t i (h ; M )

=

with the property that

= A. We have k = M = 3 since player 2 selects her strategy L 11

only when coalition f1; 2g plays (U; L). Player 3 selects her strategy r only when coalition f1; 3g plays (U; r). A third period is needed for player 1 to play D.

6

Concluding comment

We have introduced the concept of minimal strong curb sets which extends the notion of curb sets by allowing for group deviations. Similarly to strong curb sets, we can de…ne the notion of strong prep sets. Strong prep sets are product sets of pure strategies such that each player’s set of recommended strategies contains at least one action she may rationally select in every coalition she might belong to, for any belief each coalition member may have that is consistent with the recommendations to the other players.6

Acknowledgements We thank the editor, an anonymous referee, Jean-François Mertens, Olivier Tercieux and Jorgen Weibull for useful comments and suggestions. Ana Mauleon and Vincent Vannetelbosch are Senior Research Associates of the National Fund for Scienti…c Research (FNRS), Belgium. Financial support from the Spanish Ministry of Economy and Competition under the project ECO2012-35820 and from the Fonds de la Recherche Scienti…que - FNRS research grant J.007315 are gratefully acknowledged.

Appendix We now show that the existence of minimal strong curb sets holds in general. Let G

be the class of normal-form games G = N; fAi gi2N ; fui gi2N where for each player

i 2 N = f1; 2; : : : ; ng, Ai is a compact subset of a metric space and ui : A ! R

is a continuous von Neumann-Morgenstern utility function. Payo¤s are extended 6

Prep sets (Voorneveld, 2004) are product sets of pure strategies such that each player’s set of

recommended strategies must contain at least one best-response to whatever belief she may have that is consistent with the recommendations to the other players. Voorneveld (2005) has shown that, in generic games, persistent retracts (Kalai and Samet, 1984), minimal prep sets and minimal curb sets coincide. See also Kets and Voorneveld (2008).

12

to mixed strategies in the usual way. Let measures over Ai . If Bi

(Ai ) be the set of Borel probability

Ai is a Borel set, then

probability measures with support in Bi :

(Bi ) denotes the set of Borel

(Bi ) = f

i

2

(Ai ) j

i (Bi )

= 1g. If

G 2 G, that is, payo¤ functions are continuous and strategy sets compact, then each set BRi (

X =

Ai is nonempty and compact. A strong curb set is a product set

i)

i2N Xi

where (a) for each i 2 N , Xi

pure strategies; (b) 8J J

N; 8

j2J Xj .

CBR ( )

=(

1 ; :::;

Ai is a nonempty, compact set of n)

with

i

2

(Xl ), i 2 N ,

l2N nfig

Theorem 1. Every game G 2 G has a minimal strong curb set. Proof. Let Q = strong-curb(G) denote the collection of all strong curb sets of G. A is a strong curb set of G since for every J with

i

2

l2N nfig

N and

(Al ), i 2 N , we have CBRJ ( )

= (

j2J Aj .

1 ; :::;

N)

So Q is non-

empty and partially ordered via set inclusion. According to the Hausdor¤ Maximality Principle, Q contains a maximal nested subset R. For each i 2 N , let

Xi = \Y 2R Yi be the intersection of player i’s strategies in the nested set R. The set Xi is nonempty since the conditions of the Cantor intersection principle are satis…ed, i.e. (i) the collection fYi j Y 2 Rg is nested and thus satis…es the …-

nite intersection property and (ii) each Yi is nonempty and compact. It remains to prove that X =

with

i

for J

2

i2N Xi

l2N nfig

is a minimal strong curb set. Take

(Xl ), i 2 N . We have that CBR ( ) \ J

N since CBR ( ) \

CBRJ ( ) \ ([Y 2R j2J (Aj nYj )

=(

J

j2J

j2J (Aj nXj )

J

= CBR ( ) \ (

(Aj nYj )) = [Y 2R (CBRJ ( ) \

j2J

1 ; :::;

j2J (Aj nXj )

N)

= ;

[Y 2R (Aj nYj ))

j2J (Aj nYj ))

and CBRJ ( ) \

= ; for all Y 2 R (Y is a strong curb set). This establishes that X is

a strong curb set. The fact that it is minimal follows directly from the fact that R is a maximal nested subset of Q.

References [1] Ambrus, A., "Coalitional rationalizability," Quarterly Journal of Economics 121, 903-929 (2006). [2] Ambrus, A., "Theories of coalitional rationality," Journal of Economic Theory 144, 676-695 (2009).

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[3] Aumann, R.J., "Acceptable points in general cooperative n-person games," in Contributions to the theory of games IV, Princeton University Press, pp.287-324 (1959). [4] Basu, K. and J.W. Weibull, "Strategy subsets closed under rational behavior," Economic Letters 36, 141-146 (1991). [5] Bernheim, B.D., B. Peleg and M. D. Whinston, "Coalition-proof Nash equilibria: I. Concepts," Journal of Economic Theory 42,1-12 (1987). [6] Herings, P.J.J., A. Mauleon and V. Vannetelbosch, "Rationalizability for social environments," Games and Economic Behavior 49, 135-156 (2004). [7] Hurkens, S., "Learning by forgetful players," Games and Economic Behavior 11, 304-329 (1995). [8] Kalai, E. and D. Samet, "Persistent equilibria in strategic games," International Journal of Game Theory 14, 41-50 (1984). [9] Kets, W. and M. Voorneveld, "Learning to be prepared," International Journal of Game Theory 37, 333-352 (2008). [10] Luo, X. and C. Yang, "Bayesian Coalitional Rationlizability," Journal of Economic Theory 144, 248-263 (2009). [11] Voorneveld, M., "Preparation," Games and Economic Behavior 48, 403-414 (2004). [12] Voorneveld, M., "Persistent retracts and preparation," Games and Economic Behavior 51, 228-232 (2005).

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